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182 ALGEBRA

2. If A and B are square matrices of the same size and C = AB,then
–1
–1
–1
C = AB or S (AB)S =(S AS)(S BS).
2
2 2
3. If A is a square matrix and C = λA where λ is a scalar, then
–1
–1
C = λB or S (λB)T = λS BS.
2
2
4. Two similar matrices have the same rank, the same trace, and the same determinant.
Under some additional conditions, there exists a similarity transformation that turns a
square matrix A into a diagonal matrix with the eigenvalues of A (see Paragraph 5.2.3-5) on
the main diagonal. There are three cases in which a matrix can be reduced to diagonal form:
1. All eigenvalues of A are mutually distinct (see Paragraph 5.2.3-5).
2. The defects of the matrices A–λ i I are equal to the multiplicities m of the corresponding

i
eigenvalues λ i (see Paragraph 5.2.3-6). In this case, one says that the matrix has a simple
structure.
3. Symmetric matrices.
For a matrix of general structure, one can only find a similarity transformation that
reduces the matrix to the so-called quasidiagonal canonical form or the canonical Jordan
form with a quasidiagonal structure. The main diagonal of the latter matrix consists of the
eigenvalues of A, each repeated according to its multiplicity. The entries just above the
main diagonal are equal either to 1 or 0. The other entries of the matrix are all equal to zero.
The matrix in canonical Jordan form is a diagonal block matrix whose blocks form its main
diagonal, each block being either a diagonal matrix or a so-called Jordan cell of the form
⎛ 1 0 0 ⎞
λ k ···
0 λ k 1 ··· 0
0 0 0
⎜ ⎟
⎟
⎜ λ k ··· ⎟ .
⎝ . . . . . . . . . . . . . . . ⎠
Λ k ≡ ⎜
0 0 0 ··· λ k
5.2.3-3. Congruent and orthogonal transformations.
Square matrices A and A of the same size are said to be congruent if there is a nondegenerate
2
square matrix S such that A and A are related by the so-called congruent or congruence
2
transformation
T
T
A = S AS or A = SAS .
2
2
This transformation is characterized by the fact that it preserves the symmetry of the
original matrix.
For any symmetric matrix A of rank r there is a congruent transformation that reduces
A to a canonical form which is a diagonal matrix of the form,
( )
I p
T
A = S AS = –I r–p ,
2
O
where I p and I r–p are unit matrices of size p × p and (r – p) × (r – p). The number p is called
the index of the matrix A,and s = p –(r – p)= 2p – r is called its signature.
THEOREM. Two symmetric matrices are congruent if they are of the same rank and have
the same index (or signature).
–1
T
A similarity transformation defined by an orthogonal matrix S (i.e., S = S )is said
to be orthogonal. In this case
–1
T
A = S AS = S AS.
2